TR-2016-01

Chip-firing based methods in the Riemann-Roch theory of directed graphs

Abstract

Baker and Norine proved a Riemann-Roch theorem for divisors on undirected graphs.
The notions of graph divisor theory are in duality with the notions of the chip-firing game.
Based on this connection, we give a new proof for the Riemann-Roch theorem on graphs which
can be generalized to Eulerian directed graphs, improving a result of Amini and Manjunath.
We also give a graph-theoretic version of the abstract Riemann-Roch criterion of Baker and Norine,
and explore the natural Riemann-Roch property introduced by Asadi and Backman.